**Bernhard Heim**

German University of Technology, Oman

September 27, 2012

**On a hidden property of Borcherds products**:
Borcherds products are special automorphic forms for the orthogonal
group $O_{2,n}(\R)$ of signature $(2,n)$, lifts from weakly modular
forms. To state explicit examples we identify $O_{2,2}$ with $SL_2
\times SL_2$ and $O_{2,3}$ with the Siegel modular group of degree
$2$.
Let $j$ be the modular invariant with Fourier coefficients $c(n)$,
normalized with $c(0)=0$. Then one of the most famous examples is
given by $$j(z)-j(w) = p^{-1} \prod_{m=1, n\geq -1}^{\infty} \left( 1 - p^m q^n\right)^{c(nm)} \quad \left(p:=e^{2 \pi i z}, q:=e^{2 \pi i w}\right), $$
related to the denominator formula of the monster Lie algebra. This
formula had been known before by Koike, Zagier. New was the link to
the Moonshine conjecture and a systematic construction of automorphic products called Borcherds lifts due to Borcherds.
The case of Siegel modular forms has been first studied by Gritsenko and Nikulin. They found a deep connection between the Igusa function
$\Delta_5$ and a certain Kac-Moody algebr